The prefix-free Kolmogorov complexity K was defined on strings but it is also common to write K(n) for numbers n. Here we identify as usual the nth string σn in the length-lexicographic ordering of all strings with the number n, and so we have K(n) = K(σn).
However, there are several ways to define K(n), depending on the way one wishes to represent numbers by strings. Another possibility is for ex-ample to consider K(0n). The two possibilities do not differ substantially since
K(n) ≤ K(0n) + O(1) ≤ K(n) + O(1), (4.3) and we do not care about additive constants here. For other representations one gets similar relations which is a consequence of the following well-known lemma.
Lemma 4.10. If g is a computable function from strings to strings then K(g(σ)) ≤ K(σ) + O(1) for each string σ.
Remark 4.11. It is easy to verify that the following analog of Lemma 4.10 for computable machines is true: If g is a computable function from strings to strings and if M is a computable machine, then there is a computable machine N such that KN(g(σ)) ≤ KM(σ)+O(1) for each string σ. Hence the choice of the representation of numbers does not matter in the computable machine setting, either. For further reference, we mention that in particular we have the following analog of (4.3).
For each computable machine M there is a computable machine N such that (∀n ∈ ω) KN(n) ≤ KM(0n) + O(1). (4.4)
C By definition, a sequence X is K-trivial if for all n, K(X n) ≤ K(n) + O(1). I.e., all initial segments of X have minimal prefix-free complexity (within an additive constant). The class of K-trivials was introduced by Chaitin [9]. A central set of results in the theory of algorithmic randomness were proved by Nies and Hirschfeldt. They show the equivalence of a number of “anti-randomness” properties, including the property of being K-trivial.
To define another such property called lowness for Martin-L¨of randomness, one considers Martin-L¨of tests relative to an oracle. Namely, a sequence of sets A0, A1, . . . is called a Martin-L¨of test relative to an oracle X if there is a computable function g such that for each n, An= Wg(n)X and µ [An] ≤ 2−n. Further, a sequence Y is Martin-L¨of random relative to an oracle X if Y
withstands every Martin-L¨of test relative to X, i.e., if for for every Martin-L¨of test AX0 , AX1 , . . . relative to X, we have that Y 6∈ ∩nAXn. Now a sequence X is called low for Martin-L¨of randomness if each Martin-L¨of random sequence is Martin-L¨of random relative to X.
Theorem 4.12 (Nies and Hirschfeldt [34]). For every sequence X ∈ 2ω, the following are equivalent:
– X is low for K.
– X is K-trivial.
– X is low for Martin-L¨of randomness.
We briefly mention K-reducibility. A sequence X is K-reducible to a sequence Y if for all n, K(X n) ≤ K(Y n) + O(1); in this case we write X ≤K Y . Then X is K-trivial iff X ≤K0ω.
An analog of K-triviality in the case of Schnorr randomness was in-troduced by Downey and Griffiths [13]. They define Schnorr reducibility as follows. A sequence X is Schnorr reducible to a sequence Y if for ev-ery computable machine M there is a computable machine N such that KN(X n) ≤ KM(Y n) + O(1). Now a sequence X is Schnorr trivial if X is Schnorr reducible to 0ω. For results on Schnorr triviality see [12, 13, 19].
We study an analog triviality concept based on bounded machines.
Definition 4.13. (i) A sequence X is boundedly reducible to a sequence Y if for every bounded machine M there is a bounded machine N such that
(∃c ∈ ω)(∀n ∈ ω)KN(X n) ≤ KM(Y n) + c.
In this case we write X ≤bndY .
(ii) A sequence X is boundedly trivial if X ≤bnd0ω.
As the following proposition shows, there is a characterization of bounded triviality where the prefix-free Kolmogorov complexity K is involved. As for the proof, we recall that the letter U denotes the universal prefix-free ma-chine relative to which the prefix-free Kolmogorov complexity K = KU is defined.
Proposition 4.14. (i) There is a bounded machine Ub such that KUb(0n) ≤ K(n) + O(1).
(ii) A sequence X is boundedly trivial if and only if there is a bounded machine N such that
(∃c ∈ ω)(∀n ∈ ω)KN(X n) ≤ K(n) + c.
Proof. Clearly, (ii) follows from (i). To prove (i), we define a prefix-free machine Ub by letting Ub(τ ) = 0|U (τ )| for all τ ∈ dom(U ). Obviously, Ub
is bounded via the probability measure ν which is given by ν(0n) = 1 for each n. By construction, we have KUb(0n) ≤ K(0n) ≤ K(n) + O(1) for all n.
Corollary 4.15. Every computable sequence is boundedly trivial.
Proof. Suppose X is computable. Similar to the proof of Proposition 4.14, we define a bounded machine N as follows. For all τ ∈ dom(U ), determine the number n such that U (τ ) is the nth string in 2<ω, and let N (τ ) = X n.
Clearly, N is bounded via the probability distribution ν given by ν(σ) = 1 for all σ ≺ X, and we have KN(X n) ≤ K(n) + O(1).
The next corollary follows from Proposition 4.14 (ii) but will also be an immediate consequence of Theorem 4.19.
Corollary 4.16. Every boundedly trivial sequence is K-trivial.
Remark 4.17. There exists no bounded machine M such that KM(n) ≤ K(n) + O(1).
Otherwise, by Theorem 4.5 we would have that a sequence is computably random if and only if it is Martin-L¨of random. Hence there is no bounded machine N such that KN(n) ≤ KUb(0n)+O(1) for all n, where Ubdenotes the bounded machine from Proposition 4.14 (i). It follows that an analog of (4.4) is not true for bounded machines, and therefore an analog of Lemma 4.10 for bounded machines is also false (see Remark 4.11). C
The above remark suggests the following definition.
Definition 4.18. A sequence X is weakly boundedly trivial if for every bounded machine M there is a bounded machine N such that
(∃c ∈ ω)(∀n ∈ ω)KN(X n) ≤ KM(n) + c.
In the following theorem, an interesting relation to the K-trivial se-quences is established.
Theorem 4.19. (i) Every boundedly trivial sequence is weakly boundedly trivial.
(ii) Every weakly boundedly trivial sequence is K-trivial.
Lemma 4.20 (folklore). For every sequence X, if a0, a1, . . . is an increas-ing and unbounded computable sequence of numbers, and if K(X an) ≤ K(an) + O(1), then X is K-trivial.
Proof. We define a prefix-free machine M as follows. If M finds a τ ∈ dom(U ), it checks whether there is an index n such that an = |U (τ )|. If there exists such an n, which by hypothesis is not greater than an, the machine M outputs U (τ ) n. It follows that for all n
K(X n) ≤ K(X an) + O(1) ≤ K(an) + O(1) ≤ K(n) + O(1).
Proof of Theorem 4.19. Item (i) is an immediate consequence of Proposi-tion 4.14 (ii). We prove item (ii). Considering our standard representaProposi-tion of numbers by strings, where the nth string σn in the length-lexicographic ordering is identified with the number n, we let a0, a1, a2, a3. . . denote the numbers that correspond to the strings λ, 0, 02, 03. . .. Now let Ub de-note the machine from Proposition 4.14 (i). If X is a weakly boundedly trivial sequence, then there is a bounded machine N such that for all n, KN(X n) ≤ KUb(n) + O(1). So we have
K(X an) ≤ KN(X an) + O(1) ≤ KUb(an) + O(1) ≤ K(an) + O(1).
Hence by Lemma 4.20, X is K-trivial.
Chapter 5
Schnorr Randomness:
Lowness for Computable Machines
By Theorems 4.5 and 2.30, computably random sequences and Schnorr ran-dom sequences can be characterized w.r.t. their initial segment complexities via bounded and computable machines, respectively. While in Section 4.2, a lowness notion for bounded machines was investigated, we define in this chapter a lowness notion for computable machines. We show that the se-quences which are low for computable machines are exactly the computably traceable sequences. Thus by known results, lowness for computable ma-chines is equivalent to other lowness notions with respect to Schnorr ran-domness. Namely, a sequence X is low for computable machines iff X is low for Schnorr tests iff X is low for Schnorr randomness. The latter two proper-ties and the definition of computable traceability are reviewed in Section 1, where we also introduce lowness for computable machines. In Section 2, we prove the above mentioned coincidence of the class of sequences which are low for computable machines and the class of computably traceable sequences.
5.1 Lowness Notions for Schnorr Randomness
Below Remark 4.11, we reviewed the definition of lowness for Martin-L¨of randomness via relativizations of Martin-L¨of tests and of Martin-L¨of ran-domness. In what follows, we recapitulate similar lowness notions for Schnorr randomness, and we introduce lowness for computable machines.
In the following definition we use a straightforward relativized version of uniformly computable sequences of reals (see Definition 2.21).
Definition 5.1. Let X ∈ 2ω.
(i) A Schnorr test relative to X is a Martin-L¨of test AX0 , AX1 , . . . rela-tive to X such that µAX0 , µ AX1 . . . is a uniformly X-computable sequence of reals.
(ii) A sequence is Schnorr random relative to X if it withstands every Schnorr test relative to X.
Definition 5.2. A sequence X is low for Schnorr tests if for every Schnorr test AX0 , AX1 . . . relative to X there is a Schnorr test B0, B1, . . . such that
∩nAXn ⊆ ∩n[Bn].
Recall from Subsection 1.2 that for each n ∈ ω, Dn denotes the finite set whose canonical index is n.
Definition 5.3 (Terwijn and Zambella [45]). A sequence X ∈ 2ω is com-putably traceable if there is a computable function h such that the following condition is satisfied. For all functions g ≤T X, there is a computable func-tion r such that for the finite sets Dr(n) we have that |Dr(n)| ≤ h(n) and g(n) ∈ Dr(n).
We remark that, as noticed by Terwijn and Zambella, if X is computably traceable then for the witnessing function h we can choose any computable, nondecreasing and unbounded function.
Theorem 5.4 (Terwijn and Zambella [45]). A sequence X is low for Schnorr tests if and only if X is computably traceable.
We note that while all K-trivials are ∆02 by a result of Chaitin [9], the computably traceable sequences are all of hyperimmune-free degree, and there are 2ℵ0 many of them.
Definition 5.5. A sequence X is low for Schnorr randomness if each Schnorr random sequence is Schnorr random relative to X.
It is not hard to see that if a sequence is low for Schnorr tests then it is also low for Schnorr randomness. Whether the converse also holds was an open question of Ambos-Spies and Kuˇcera [1]. It was answered in the affirmative by Kjos-Hanssen, Nies, and Stephan.
Theorem 5.6 (Kjos-Hanssen, Nies, and Stephan [23]). A sequence is low for Schnorr randomness if and only if it is low for Schnorr tests.
Note that if we consider an analog notion of lowness for Martin-L¨of tests then the equivalence of that notion to lowness for Martin-L¨of randomness is easy to prove. One direction is trivial as above in the Schnorr case. The other direction is a consequence of the following fact which is an analog of Theorem 2.3: Given some sequence X there is a Martin-L¨of test U0X, U1X, . . . relative to X which is universal for all Martin-L¨of tests relative to X, i.e., if AX0 , AX1 , . . . is a Martin-L¨of test relative to X then ∩nAXn ⊆ ∩nUnX.
We shall introduce another lowness notion for Schnorr randomness which is an analog of K-triviality in the setting of Martin-L¨of randomness. For any X ∈ 2ω, an X-computable machine is a prefix-free Turing machine M such that µ [dom M ] is an X-computable real.
Definition 5.7. A sequence X ∈ 2ω is low for computable machines if for all X-computable machines M there is a computable machine N such that for all n,
KN(n) ≤ KXM(n) + O(1).
Similar to the reasoning before Proposition 4.7, we can argue that an X-computable machine M need not be Y -X-computable for all oracles Y . How-ever we have the following proposition, the proof of which uses a straight-forward relativized version of Remark 2.31.
Proposition 5.8. For every sequence X and for every X-computable ma-chine M there is an oracle Turing mama-chine fM such that MX ' fMX and M is a Y -computable machine for every oracle Y .f
Proof sketch. Let X be a sequence and let M = Me be an X-computable machine. We may assume that for every oracle Y , MY is prefix-free, i.e., WeY is a prefix-free set. We define the oracle Turing machine fM as follows.
Let F : 2ω× ω → ω be a partial computable functional with F (Y, 0) = 0 for all Y ∈ 2ω such that for every n > 0, F (X, n) is defined and greater than F (X, n − 1), and
µdom MX − µh
We,F (X,n)X i
≤ 2−n.
Now for any oracle Y , the machine fMY on input x executes the following instructions. At stage n > 0, first wait for F (Y, n) to converge. If F (Y, n) <
F (Y, n − 1) then let the computation diverge. Else continue as follows. If the condition
(∀m < n) µh
We,F (Y,n)Y i
− µh
We,F (Y,m)Y i
≤ 2−m (5.1)
is satisfied then do the following: If x ∈ We,F (Y,n)Y then output MY(x) and stop, else move to stage n + 1. (Note that x < F (Y, n) whenever x ∈ We,F (Y,n)Y .) On the other hand, if (5.1) is not satisfied then let the computation diverge. Note that the construction is uniform in M, F but not in M alone.
Recall that a sequence X is Schnorr trivial if for every computable machine M there is a computable machine N such that for all n,
KN(X n) ≤ KM(n) + O(1)
(see page 43). This notion was initially explored by Downey and Grif-fiths [13] and Downey, GrifGrif-fiths and LaForte [12], who showed that this class does not coincide with the sequences that are low for Schnorr random-ness. For instance, there are Turing complete Schnorr trivial sequences.
In the next section we show that unlike the situation for triviality, the coincidence of the sequences low for Martin-L¨of randomness and the low for K ones carries over to the Schnorr case (cf. Theorem 4.12).